Physics

Problem

An object oscillates in simple harmonic motion along a straight line. The acceleration–time graph for the motion is described below. • The acceleration varies sinusoidally with period

0.80\,\text{s}

. • The minimum value on the graph is

-2.0\,\text{m\,s}^{-2}

t = 0

. • The graph crosses zero acceleration for the first time at

t = 0.20\,\text{s}

. Use this information to determine key dynamic quantities of the motion. (a) Determine the frequency

f

of the oscillation. (b) Calculate the amplitude

A

of the displacement. (c) Compute the object’s maximum speed

v_{\text{max}}

Intuitive explanation

In simple harmonic motion, acceleration and position are directly linked: $a = -\omega^2 x$ . The acceleration graph tells you the angular frequency $\omega$ (from the period) and the amplitude $A$ (from the maximum acceleration magnitude). Since you know when $a$ reaches its extreme and crosses zero, you can reconstruct the full sinusoidal equation for $a(t)$ , then use the relationship between $a$ , $v$ , and $x$ to find velocity and displacement. The key is recognizing that the acceleration graph's features—its period, extrema, and zero-crossings—encode all the information about the oscillator's size and speed.

Intuitive explanation

Browse the problem library